I think the mathematicians are saying there are many more dimensions than just three or four. I don't recall now the number, but more than 7(??).

Your analogy of the balloon is our best picture of it yet, even if we can't visualize four dimensions (since we're only 3D creatures). You may have answered your own question, however. There is "nothing" outside the universe, so we may be expanding into nothing. The question presumes that there is "something" to expand into. I don't know that we know that. There may not be an "outside" of the universe.

Lawrence Krauss presented a reasonably good case that the universe came from nothing, and will go to nothing, because the net energy of the universe is zero. I'm not sure of the status of that hypothesis in the peer-reviewed community.

There is also the "frothy" multiverse hypothesis, where independent universes simply expand and grow, like the bubbles in a beer, totally unaware of the other universes. I don't know how we'd ever prove that hypothesis. I assume the mathematics work out at some level, or it never would have been suggested.

I was a physicist, but not a cosmologist. So I only know the bits and pieces from what I've read, and occasionally discussed with others. I did read Krauss' book, which I found pretty accessible (i.e. you don't need to be a physicist to read it). Though I don't recall him specifically dealing with what we may (or may not) be expanding "into".

Fun to think about this sort of thing sometimes.